Let's solve the equation step-by-step:
Given:
[tex]\[\log_2 4 + \log_2 4 + \log_2 x = 6\][/tex]
1. Simplify the logarithmic terms:
We know that:
[tex]\[\log_2 4 = 2\][/tex]
because [tex]\(2^2 = 4\)[/tex].
So, we can replace [tex]\(\log_2 4\)[/tex] with 2:
[tex]\[2 + 2 + \log_2 x = 6\][/tex]
2. Combine the constants:
Adding the constant terms:
[tex]\[2 + 2 = 4\][/tex]
So now the equation becomes:
[tex]\[4 + \log_2 x = 6\][/tex]
3. Isolate the logarithmic term:
To isolate [tex]\(\log_2 x\)[/tex], subtract 4 from both sides of the equation:
[tex]\[\log_2 x = 6 - 4\][/tex]
4. Simplify the right side:
[tex]\[\log_2 x = 2\][/tex]
5. Rewrite the logarithmic equation in exponential form:
By definition of logarithms, if [tex]\(\log_b y = z\)[/tex], then [tex]\(b^z = y\)[/tex]. Thus:
[tex]\[2^2 = x\][/tex]
6. Calculate the value:
[tex]\(2^2 = 4\)[/tex].
Therefore, the value of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
